Looking at various recent examination papers, it has become clear to
me that there is significant confusion between these two questions.
This post is intended to bring some clarity to the situation.

At the start of this post, I will give an example of the confusion as
it appears in exam questions (and probably elsewhere), and clarify
what the two different phrases mean using the above example. I will
then delve more deeply into the mathematics of these two things, going
beyond A-level content, and use some undergraduate analysis to find
equivalent conditions for them in terms of the derivatives of the
functions. It is fine to skip over the technical stuff and just look
at the results (theorems)!

(Exactly the same applies to the use of the term “decreasing”, but for
simplicity we will focus on increasing functions in this post.)

Having recently listened to about 5.5 hours of Craig
Barton interviewing Dani Quinn (part
1
and part
2),
the Head of Mathematics at Michaela Community
School, I decided that it was worth visiting
the school to see their principles in action for myself, so last week,
I took to the buses to visit Wembley.

I thought a bit more about my previous
post on small
angle approximations, and decided it might be helpful to describe an
application of the small angle approximations. While this example
contains non-examinable aspects (at least in single maths A-level),
the context should be fairly familiar (or can easily be demonstrated),
and the mathematics is accessible to single maths students (at least
as a demonstration). It also ties together ideas from mechanics and
pure maths, so is helpful in this regard.